RU2625642C1 - Method of simultaneous calibration of three and more single-different devices with measuring functions without support to the reference device or reference test signal - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: calibration method is based on the use of two or more additional devices of the same type with a calibrated device. In this case, all devices are equal. As an indicator of the accuracy of each device, the standard deviation (SD) of its random component of error is taken. It is assumed that this component of error exceeds the other components of the error. When implementing the proposed method, the discrete readings of the calibrated device are compared with the readings of additional devices, and the value of the accuracy factor for each device is evaluated and the variances of the obtained values are calculated. In this case, the accuracy of each device has the property of unbiasedness.

EFFECT: increased application area, improved calibration accuracy.

3 dwg

Description

The invention relates to metrology, instrumentation, navigation instrumentation and other areas that use in one form or another measuring equipment, in particular to methods for calibrating measuring instruments (hereinafter - SI) in order to evaluate their accuracy indicators, where the terms "calibration", "SI" and “accuracy indicator” are defined in [1]. Moreover, the SI calibration procedure is considered in this invention not in full, but only in terms of quantitative assessment of its accuracy characteristics, including with the aim of confirming their compliance with the established requirements, not including the task of metrological traceability of measurement results to the standard [1].

In addition to SI, the invention can be applied to any technical systems and devices with measuring functions (hereinafter - TSUIF [2]), including the case of inadmissibility of dismantling their measuring part (for calibration), if it can lead to violation of the target functions of these systems. The invention can also be applied in the field of technological preparation and support of various industries using SI and TSUIF if it is necessary to carry out their certification and conformity confirmation procedures, including for cases of direct use of SI and TSUIF (during the specified maintenance) and when they are included existing automated process control systems. The invention can also be applied to tests of finished products containing TSUIF in order to confirm the conformity of the characteristics of their measuring part with the established requirements. The invention also makes it possible to perform self-diagnosis of measuring components (if any) of various systems, including autonomous systems.

Examples of such SI and TSUIF (hereinafter referred to as devices) can serve as such devices as a direction indicator and a lag (which are part of the inertial navigation system), a wave-measuring buoy (recording the current parameters of sea waves as part of the stabilization system of the offshore platform), as well as a number of other devices for which is relevant to the local calibration task, consisting in evaluating the indicators of their real accuracy. To solve this problem, the functional representation of the device is adopted in the form shown in FIG. one.

In the indicated FIG. 1 marked:

- “object” - an element of some external (with respect to the device) system of natural or artificial origin, which must be controlled (or studied) using the device;

- Q (t) - a variable input signal acting on the sensitive element of the device, which is generated by the object factor of interest to us and which is aligned with the measured physical quantity, where the latter is traditionally understood as some parameter of the accepted model representation of the object [1];

- S (t) is the output signal (indication) of the device, taken as the measurement result;

- t is the current time.

Moreover, the accuracy of the device can be characterized by various parameters of its error ξ (t) = S (t) -Q (t) [3]. The indicated error parameters are specified below and are subject to evaluation during calibration, which must be performed in order to justify the possibility of using the measurement result S (t) for its intended purpose.

We specify a number of aspects of devices calibrated in accordance with this invention. First of all, it is assumed that each of these devices has the function of recording on a magnetic medium (in a file on a disk) its discrete readings S _i = S (t _i ) at times t _i = t ₀ + (i-1) ⋅Δt, the following with a certain sampling period Δt, where t ₀ is the recording start time, i = 1,2, ..., n is the reading number, n is the length of the recorded implementation. Moreover, taking into account that the error ξ (t), which is burdened by the output signal S (t), has a random component, it is assumed that the sampling period Δt is chosen no less than the autocorrelation time of the output signal S (t) for each of these devices. It is also assumed that data recording from all devices is performed synchronously, including with the same sampling period Δt. For the purposes of the further discussion, we also introduce the abbreviated notation Q _i = Q (t _i ), ξ _i = ξ (t _i ), similar to S _i .

In addition, we believe that in accordance with [1, 2], the design and technological development during the creation of each of the devices under consideration was performed at the proper, fairly high level. In particular, it is assumed that all additional factors affecting the output signal S (t) are identified and studied, after which either these factors are stabilized or the corresponding systematic distortions of the output signal S (t) are automatically compensated at the hardware or software level. These factors include, first of all, climatic, mechanical, electromagnetic factors (temperature, pressure, humidity, vibration level, power supply noise, etc.), as well as a number of other more specific factors, such as pitching and movement parameters marine objects for the above device examples.

Under these assumptions, the error ξ (t) of each of the considered devices can be described by a purely stochastic stationary model. Moreover, to maintain the stationarity of the indicated model, the following general assumptions are additionally made with respect to the input signal Q (t):

(a) Q (t) ∈ [Q _min , Q _max ] is the limited range of possible signal values;

(b) ⎪dQ (t) / dt⎪ << ΔQ / Δt is the smoothness of the signal change, where ΔQ = Q _max -Q _min .

This allows us to neglect the dynamic component of the device error due to sharp changes in the input signal Q (t), and to consider the stochastic model of the error ξ (t) stationary over the entire time the object is observed. In addition, the assumption of ergodicity [4] of the error model, which is usual in such cases, is used, which allows replacing the averaging over the ensemble of output signals S (t) with the averaging over time of one implementation of the indicated signal.

Based on the foregoing, in the present invention, the following model of the discrete output signal {S _i } (i = 1,2, ..., n) of each device is assumed:

where Q _i is the current value of the input signal (measured value), and the errors {ξ _i } are assumed to be random variables that are equally distributed with the first two constant moments { a , σ ² } and pairwise uncorrelated: cov (ξ _i , ξ _j ) = 0 for i ≠ j. In this model representation, the indicated parameters are considered unknown and are understood as the residual zero shift of the device a and the standard deviation σ (hereinafter - RMS) of its random error. The specified parameters receive their specific values in the course of the necessary preliminary adjustment of each device in accordance with [1], as well as the procedures for its inclusion and exit to operating mode. Adjusting the device in the simplest case consists in eliminating the zero offset of its readings (to the maximum extent possible) and refining the gain in the entire range of the device readings, and taking into account what was said above about compensation for the main systematic distortions of the output signal, we can consider the condition | and | << σ.

гарантированного качества, где степень качества оценки, как обычно, характеризуется двумя параметрами: ее смещением

и дисперсией

[5]. Здесь и далее E(…) и D(…) обозначают, соответственно, операторы математического ожидания и дисперсии, применяемые к случайным величинам, в том числе, и к оценкам, вычисляемым по экспериментальным данным. Кроме того, в обоснованных случаях вместо указанных параметров

и

используются их безразмерные аналоги, обозначаемые ниже, как

и

.Thus, in the conditions described above, the main indicator of the accuracy of the device is the parameter σ - the standard deviation of the random component of its error, and the task of calibrating the device is understood in this invention as obtaining an estimate

guaranteed quality, where the degree of evaluation quality, as usual, is characterized by two parameters: its bias

and dispersion

[5]. Hereinafter, E (...) and D (...) denote, respectively, the mathematical expectation and variance operators applied to random variables, including estimates calculated from experimental data. In addition, in justified cases, instead of the specified parameters

and

their dimensionless counterparts are used, denoted below as

and

.

In particular, if the device to be calibrated is a SI, for which tests were carried out in the prescribed manner with the aim of confirming the type with subsequent inclusion in the State Register of SI [6], then its calibration in the above assumptions about the device and in terms of evaluating its accuracy characteristics by content will practically coincide with the verification procedure of this SI, differing only in the legal status of the results.

Known methods and procedures for calibration (verification) of devices of this type are described in a number of monographs, manuals and methodological materials on metrology, including regulatory documents [7-9]. At the same time, despite some differences and varying degrees of detail in the description of these procedures, all known sources suggest the mandatory use of a model (reference) pre-certified device whose accuracy is not only assumed to be known (or normalized, for example, by its upper maximum permissible limit of standard deviation σ _max ), but also significantly exceeding the expected accuracy of the calibrated device. This requirement significantly limits the scope of known calibration methods, which, as a rule, must be performed in certified laboratories with metrological status, as well as with the involvement of specialized devices and equipment.

The closest in technical essence and adopted for the prototype is a known method of calibrating the device [10, §56]. Also known details of the prototype method for the case under consideration is the dominance of a random error [11, section 10-2]. Although formally the prototype method [10, 11] describes a method for verifying SI, it can obviously be extended to the calibration method (in the above sense) of the measuring part of any TSUIF. The prototype method is based on the use, in addition to the calibrated device (hereinafter - KU), of another additional device (of the same type with KU by the input signal), which is given the status of a model or reference device (hereinafter - EC). The essence of the prototype method is presented in FIG. 2, explaining that the input signal Q (t) is supplied simultaneously to the inputs of both devices KU and EI, followed by a comparison of their readings S ^K (t) and S ^E (t), where the superscripts indicate belonging to the corresponding device. This explanation also applies to additional indices in the notation of other variables used below.

In this case, the input signal Q (t) plays the role of a test one, and it is formed, as a rule, using a special generator, alternately setting the values of Q (t) in such a way as to cover all the predefined points of the KU range. In this sense, the calibration of the device according to the prototype method is an active experiment. It should also be noted that it is permissible to combine the functions of the generator and the EU in one device, which does not change the essence of the prototype method.

It is important to note that the accuracy of the EA is assumed to be known and significantly (several times) higher than the expected accuracy of the control unit, which is judged on the basis of available a priori information, for example, according to the passport data of the control unit. Thus, in the adopted notation, the basic requirement of the prototype method has the form:

where σ _E and σ _K are the standard deviations of the random error of the EI (either known or normalized) and KU (to be estimated), k is the safety factor for the accuracy of the EI. Moreover, in practice, k≈3 is chosen [10, 11], however, in a number of specific measurement areas, this coefficient lies within wider limits of 1.5 <k <10 [9].

Data processing {S _i ^K }, {S _i ^E } (i = 1,2, ..., n) is reduced to calculating a sequence of pairwise differences {ΔS _i } of readings, and then their sample dispersion D _S :

where for non-bias D _S according to [5], the coefficient 1 / (n-1) is used. Further, due to the identity relation ΔS _i = S _i ^К -S _i ^Э = ξ _i ^К -ξ _i ^Э and the uncorrelated readings of different structurally independent devices, the expression is true:

Finally, discarding the second term in (4), which is possible by virtue of requirement (2), we obtain the following estimate of the desired parameter σ _K by the prototype method:

в способе-прототипе найдена только верхняя граница ее относительного смещения

, а именно, при k≥3 установлено, что

[10, 11]. К этому следует добавить, что оценка

очевидным образом является завышенной:

. В то же время, сведения о дисперсии

, характеризующей диапазон возможного случайного разброса реализаций этой оценки, ни в способе-прототипе, ни в других известных источниках не приводятся.Regarding the analysis of the quality of this assessment

in the prototype method, only the upper boundary of its relative displacement is found

, namely, for k≥3 it is established that

[10, 11]. To this should be added that the assessment

The obvious way is overpriced:

. At the same time, dispersion information

, characterizing the range of possible random scatter of implementations of this estimate, are not given either in the prototype method or in other known sources.

The disadvantages of the prototype method are:

1) the need to use additional EA with a well-known and significantly higher (several times) accuracy compared to KU;

в части ее дисперсии;2) lack of quality assessment analysis

regarding its dispersion;

3) focus on performing an active experiment using a special generator that generates the required values of the input signal Q (t).

по способу-прототипу не выполнен полностью: показано лишь малое смещение этой оценки (при k≥3), что вовсе не гарантирует ее малого случайного разброса, который возможен и в сторону значительного занижения оценки

.Due to the disadvantages of the prototype method, its use is limited. It can be used only in metrological laboratories that monitor SR in stationary installations, the key elements of which are a certified reference (exemplary) SR and a test signal generator. As for the TSUIF, for the application of the prototype method to their measuring part, either the ability to dismantle it (without violating the target functions of the TSUIF) or the availability of a mobile version of a certified reference device are necessary; at the same time, in practice, both of them are far from always available. In addition, quality assessment analysis

the prototype method is not fully implemented: only a small bias of this estimate is shown (for k≥3), which does not guarantee its small random spread, which is also possible in the direction of a significant underestimation of the estimate

.

At the same time, there are a number of uses of SI and TSUIF with a confirmed model of output signals (1), for which an operational, on-site control of their accuracy (stability) would be extremely important regardless of the presence of a reference device. As an example, we can cite the certification and conformity assessment procedures in the field of technological preparation and maintenance of various industries in terms of SI and TSUIF, which are part of the existing automated process control systems, as well as similar procedures used in testing finished products with TSUIF in their composition and focused on checking their measuring characteristics. This also applies to the ability of various systems to carry out self-diagnostics of their measuring part (if any), and, in particular, to maintain the operability of various autonomous systems, which is necessary for the operation of space, underwater and underground (downhole) equipment, during geodetic operations carried out in remote areas, etc.

A number of metrological problems proper can be attributed to this type, a feature of which is the simultaneous study of several homogeneous SI with the aim of their certification and intercomparison in accuracy in the absence (or limited availability) of a more accurate SI. This takes place, for example, during the creation and operation of group standards at different levels, as well as during circular comparisons of national and international primary standards.

The problem that is solved by the invention is to increase the scope of the method of calibration (verification) of the devices in question.

The technical result of the proposed calibration method consists in expanding the scope of the method by not using an additional reference device having a known and higher accuracy, as well as in obtaining guaranteed estimates of the reliability of the calibration results and their increased efficiency compared to the prototype method. The proposed method also allows you to completely abandon the use of a special generator that generates the necessary values of the input signal.

The problem is solved by using, in addition to the initial control unit, the calibration of which is required to be performed initially, of two or more additional devices that are also designed to measure the same Q value, however, none of them is given a reference status, including preliminary estimates of their accuracy indicators. This means that all devices - both the original calibrated and the additional ones - are equal in calibration. In other words, the calibration procedure according to the proposed method is organized as being carried out simultaneously for all devices involved in it, for which, in connection with the foregoing, we will use the symmetric designations KU-1, KU-2, KU-3, etc. As indicators of the accuracy of these devices, as before, we take the standard deviation of the random error component of each control unit, which we denote, respectively, σ ₁ , σ ₂ , σ ₃ , etc., all of which are considered unknown and are subject to evaluation. It is important that there are no restrictions on the ratios of these parameters to each other: they can be both approximately pairwise equal and significantly differ from each other. Structurally simultaneously calibrated devices may have similarities between themselves (including being several instances of one commercially available device), or they may differ fundamentally. The only mandatory requirement for each of the devices is the conformity of the model of its output signal to the relation (1), including compliance with all aspects of the justification of this model.

It is also important that for the formation of the input signal Q (t), a special generator is not required, since the calibration by the proposed method can be performed during operational monitoring of the controlled object (see Fig. 1).

Denote by m the total number of devices involved in the simultaneous calibration. Depending on m, the present invention can be implemented in two versions:

1) the minimum (loss-free) case, with m = 3;

2) the excess case, for m> 3.

Moreover, the second modification differs from the first only a slightly more complex data processing algorithm.

For a minimal case, the essence of the proposed method is illustrated in FIG. 3, which shows that the input signal Q (t) is supplied simultaneously to the inputs of all three devices, followed by a comparison of their readings S ⁽¹⁾ (t), S ⁽²⁾ (t), S ⁽³⁾ (t).

(i=1,2,…,n; j=1,2,3), их случайные погрешности

удовлетворяют следующим условиям:

;

при i₁≠i₂ и j,k∈{1,2,3}, а также при i₁=i₂ и j≠k, где параметры

неизвестны. Обработка данных на первом этапе аналогична таковой для способа-прототипа, а именно: вычисляются последовательности разностей

показаний выходных сигналов во всех сочетаниях их пар:According to the model (1) of discrete signals

(i = 1,2, ..., n; j = 1,2,3), their random errors

satisfy the following conditions:

;

for i ₁ ≠ i ₂ and j, k∈ {1,2,3}, as well as for i ₁ = i ₂ and j ≠ k, where the parameters

unknown. The data processing at the first stage is similar to that for the prototype method, namely: sequences of differences are calculated

indications of output signals in all combinations of their pairs:

этих последовательностей:and then sample variances

of these sequences:

- средние значения разностей (6). При этом для каждой из трех выборочных дисперсий (7) остается в силе соотношение, аналогичное (4):where the indices j, k∈ {1,2,3} are chosen in accordance with (6), a

- average values of differences (6). Moreover, for each of the three sample variances (7), a relation analogous to (4) remains valid.

. Очевидным решением системы (8) являются оценки:However, unlike the prototype method, there is already not one equation (4) with two unknowns, but a system of three linear equations (8) relative to the three parameters of interest to us

. The obvious solution to system (8) are the estimates:

необходимо отметить следующие два важных аспекта. Во-первых, каждая из выборочных дисперсий (7), и, как следствие, система (8) в целом инвариантна относительно неизвестных остаточных сдвигов нулей {a _j} калибруемых устройств, даже если они не являются малыми на фоне оцениваемых СКО {σ_j}. Это существенно ослабляет исходные требования к калибруемым устройствам. Во-вторых, выборочные дисперсии (7), являющиеся правыми частями системы (8), содержат преобразованные случайные погрешности

, входящие в исходные данные

. Более того, в силу соотношений (6) выборочные дисперсии (7) являются взаимно коррелированными. Эти факты могут привести к существенному ухудшению достоверности полученных результатов (9), вплоть до потери ими физического смысла. А именно, путем моделирования по известному методу Монте-Карло [12] было получено, что при умеренной длине реализаций n<1000 имеется значимо отличная от нуля вероятность того, что наименьшая из трех оценок

окажется отрицательной. Указанная вероятность тем больше, чем меньше n и чем существеннее различие оцениваемых параметров

между собой. Таким образом, ключевым вопросом возможности применения предлагаемого способа калибровки является анализ качества оценок

, который и выполнен далее.In relation to the ratings found

The following two important aspects should be noted. First, each of the sample variances (7), and, as a consequence, system (8) is generally invariant with respect to the unknown residual shifts of zeros { a _j } of calibrated devices, even if they are not small against the background of the estimated standard deviations {σ _j } . This significantly weakens the initial requirements for calibrated devices. Secondly, the sample variances (7), which are the right-hand sides of system (8), contain transformed random errors

included in the source data

. Moreover, by virtue of relations (6), sample variances (7) are mutually correlated. These facts can lead to a significant deterioration in the reliability of the results obtained (9), up to the loss of their physical meaning. Namely, by modeling according to the well-known Monte Carlo method [12], it was found that for a moderate length of implementations n <1000 there is a significantly non-zero probability that the smallest of the three estimates

will be negative. The indicated probability is the greater, the smaller n and the more significant the difference in the estimated parameters

between themselves. Thus, the key question of the applicability of the proposed calibration method is the analysis of the quality of the estimates

, which is made further.

являются несмещенными, что следует из известного факта несмещенности выборочных дисперсий (7) [5] и линейности системы (8). Таким образом, главными и единственными параметрами качества оценок

являются их дисперсии

, характеризующие возможный случайный разброс оценок

. Оценить эти дисперсии можно одним из двух способов, в зависимости от имеющейся в наличии априорной информации о виде функций распределения погрешностей

устройств.First of all, estimates

are unbiased, which follows from the well-known fact of non-bias of sample dispersions (7) [5] and linearity of system (8). Thus, the main and only parameters of the quality of estimates

are their variances

characterizing a possible random spread of estimates

. These variances can be estimated in one of two ways, depending on the available a priori information on the form of the error distribution functions

devices.

путем прямых преобразований соотношений (6)-(9). Опуская промежуточные громоздкие выкладки, приведем только результирующие выражения:The first of the two indicated methods is applicable if it is permissible to consider these distribution functions normal, which is often the case in practice. Under these assumptions, analytical expressions were obtained for theoretical variances of estimates

by direct transformations of relations (6) - (9). Omitting the intermediate bulky calculations, we give only the resulting expressions:

, а слагаемое O(n^-2) означает величину того же порядка малости, что и n^-2. В частности, при наличии информации об ориентировочных значениях параметров

(хотя это и не предполагается обязательным в рассматриваемом изобретении) па основе (10) можно получить априорные оценки дисперсий

оценок

уже на этапе подготовки к проведению одновременной калибровки устройств. В общем же случае, после подстановки оценок

в выражения (10) и выполнения ряда тождественных преобразований, можно получить следующие апостериорные оценки

искомых дисперсий:where n is the length of the implementations

, and the term O (n ^-2 ) means the value of the same order of smallness as n ^-2 . In particular, if there is information about the approximate values of the parameters

(although this is not assumed to be mandatory in the present invention), based on (10), we can obtain a priori estimates of variances

grades

already at the stage of preparation for the simultaneous calibration of devices. In the general case, after substituting estimates

into expressions (10) and the fulfillment of a number of identical transformations, we can obtain the following posterior estimates

desired variances:

, а выборочные дисперсии

и оценки

вычислены по (7), (9). При этом на практике удобнее от оценок дисперсий

перейти к оценкам СКО

, получаемым извлечением квадратных корней из

. Одновременно с переходом к

целесообразно преобразовать выражения (11) в относительную (безразмерную) форму указанного выше вида

:where n is the length of the implementations

, and sample variances

and assessments

calculated by (7), (9). Moreover, in practice, it is more convenient from variance estimates

go to estimates of standard deviation

obtained by extracting square roots from

. Along with the transition to

it is advisable to convert expressions (11) into the relative (dimensionless) form of the above form

:

по отдельности. При этом целесообразно применять следующую ориентировочную шкалу указанных уровней:where all parameters are defined in (11). The obtained formulas (12) allow us to draw a reasonable conclusion about the confidence levels of each of the three estimates

separately. It is advisable to use the following indicative scale of the indicated levels:

, то оценка

недостоверна; необходимо увеличить n и повторить эксперимент, с контролем всех требований модели (1) и в первую очередь - условия стационарности погрешностей;1) if it turned out that

then the estimate

unreliable; it is necessary to increase n and repeat the experiment, with the control of all the requirements of the model (1) and, first of all, the conditions of stationarity of errors;

оценка

является грубым приближением СКО σ_k;2) when

assessment

is a rough approximation of the standard deviation σ _k ;

оценка

адекватна интересующему нас СКО σ_k.3) when

assessment

is adequate to the standard deviation of interest σ _k .

- а именно наименьшая из трех - окажется недостоверной, и, одновременно с этим, одна или обе из оставшихся оценок

будут признаны достоверными. Это позволяет ограничить результаты калибровки по предлагаемому способу только одним или двумя из трех устройств, если они интересуют нас в первую очередь. В то же время, как следует из (12), возможно также и одновременное повышение уровня достоверности всех трех оценок

, пропорционально

(по крайней мере, до тех пор, пока сохраняется стационарность погрешностей всех участвующих в калибровке устройств).The approximate boundary values (0.5 and 0.05) of the confidence levels given above can be changed depending on the purpose of the calibrated devices. In addition, it is possible that one of the three ratings

- namely, the smallest of the three - will turn out to be unreliable, and, at the same time, one or both of the remaining estimates

will be deemed reliable. This allows you to limit the calibration results of the proposed method to only one or two of the three devices, if they are of interest to us in the first place. At the same time, as follows from (12), it is also possible to simultaneously increase the level of reliability of all three estimates

proportionally

(at least as long as the errors of all the devices involved in the calibration remain stationary).

оценок

требует значительных объемов вычислений, и применять его следует при отсутствии априорной информации о виде указанных функций распределения, а также, если имеются основания сомневаться в их нормальности. Он использует известную процедуру «jack-knife» («складной нож») [12], относящуюся к группе методов так называемого численного ресэмплинга, и состоит в следующем. Среди множества i∈{1,2,…,n) всех номеров показаний

фиксируют небольшую долю р этих номеров i∈{i₁,i₂,…,i_[pn]}, где р∈[0,001; 0,05], а запись [pn] означает целую часть числа pn. При этом указанные номера выбирают случайным образом, и показания с этими номерами отбрасывают (блокируют). Для оставшихся троек показаний в количестве n’=n-[pn] выполняют обработку данных согласно (6)-(9), фиксируя полученные оценки

, после чего отброшенные данные возвращают обратно. Указанную процедуру со случайным прореживанием данных многократно повторяют (число повторений М>10³), каждый раз заново генерируя номера отбрасываемых показаний. Полученные варианты оценок

стандартным образом усредняют и оценивают их дисперсии (по формулам, аналогичным (7)). По этим дисперсиям уже судят о достоверности усредненных оценок

аналогично (11) и (12). При повышенных требованиях к достоверности процедуру оценивания этим способом целесообразно повторить для нескольких различных долей р временно отбрасываемых данных, чтобы убедиться в стабильности всех получаемых оценок; можно также увеличить число повторений М до 10⁴ и более.The second method for estimating variances

grades

requires significant amounts of computation, and it should be applied in the absence of a priori information about the form of the indicated distribution functions, and also if there is reason to doubt their normality. He uses the well-known procedure “jack-knife” (“folding knife”) [12], which belongs to the group of methods of the so-called numerical resampling, and consists of the following. Among the set i∈ {1,2, ..., n) of all reading numbers

fix a small fraction p of these numbers i∈ {i ₁ , i ₂ , ..., i _[pn] }, where p∈ [0,001; 0.05], and the notation [pn] means the integer part of the number pn. Moreover, these numbers are randomly selected, and readings with these numbers are discarded (blocked). For the remaining triples of readings in the amount n '= n- [pn], the data are processed according to (6) - (9), fixing the estimates obtained

after which the discarded data is returned back. The specified procedure with random data thinning is repeated many times (the number of repetitions M> 10 ³ ), each time again generating the numbers of the discarded readings. Evaluation Options Received

their variances are averaged and estimated in a standard way (using formulas similar to (7)). These variances are already judged on the reliability of the averaged estimates

similarly to (11) and (12). With increased reliability requirements, the evaluation procedure in this way is advisable to repeat for several different fractions p of temporarily discarded data to ensure the stability of all received estimates; you can also increase the number of repetitions of M to 10 ⁴ or more.

на первом этапе аналогична (6), (7) с единственным отличием, что в этом случае число

всевозможных попарных разностей выходных сигналов будет превышать их количество m. Из этого следует, что система, аналогичная (8), будет избыточной. С учетом отмеченной выше коррелированности правых частей указанной системы, известным стандартным способом ее решения является обобщенный метод наименьших квадратов (далее - ОМНК) [5]. При этом особенностью применения ОМНК (также известной) является возможность получения апостериорных оценок дисперсий получаемых оценок

, j∈{1,2,…,m), что автоматически решает проблему оценивания их достоверности.We proceed to describe the excess case m> 3 of the proposed calibration method. As already noted, in the procedural plan it does not differ from the minimum case m = 3. Namely, the input signal Q (t) is supplied simultaneously to the inputs of all m devices, followed by a comparison of their readings S ^(j) (t), j∈ {1,2, ..., m}. Processing discrete data

at the first stage, it is similar to (6), (7) with the only difference that in this case the number

all sorts of pairwise differences of the output signals will exceed their number m. It follows that a system similar to (8) will be redundant. Given the correlation of the right-hand sides of the indicated system noted above, the well-known standard method for solving it is the generalized least-squares method (hereinafter - OMNC) [5]. In this case, the peculiarity of using OMNC (also known) is the possibility of obtaining posterior estimates of the variances of the obtained estimates

, j∈ {1,2, ..., m), which automatically solves the problem of evaluating their reliability.

Based on the foregoing, the claimed combination of features allows to obtain a method for the simultaneous calibration (understood in the above sense) of three or more devices with measuring functions that are the same in measured size in the absence of a reference device or a reference test signal.

.As a first example of a specific implementation of the proposed method, we take the case of simultaneous calibration of three devices with approximately equal accuracy (for example, which are several instances of a mass-produced device) with the standard deviation ratio of their random error component σ ₁ : σ ₂ : σ ₃ ≈ 1: 1: 1. In this situation, as follows from formulas (10) - (12), we obtain satisfactory estimates of all three parameters (they will also be approximately equal) starting from n = 500, which is actually achievable in practice. It is important that the proposed calibration method does not rely on the fact of the approximate equality of the estimated standard deviations and, moreover, allows it to be checked after calibration is completed and adequate estimates are obtained

.

, k=2,3). Если же нас интересуют адекватные оценки всех трех параметров, то следует обеспечить n>2100, что также реально достижимо на практике.A second example is the simultaneous calibration of three devices with a ratio of accuracy of σ ₁ : σ ₂ : σ ₃ ≈3: 1: 1, where the first device is the main one (of interest to us first), and the other two are complementary, but which are approximately three times more precisely the main one. In this situation, satisfactory estimates of the main parameter σ ₁ can be obtained starting from n = 223 (but at the same time, estimates of the auxiliary parameters σ ₂ , σ ₃ remain very rough:

, k = 2,3). If we are interested in adequate estimates of all three parameters, then n> 2100 should be ensured, which is also achievable in practice.

Finally, as a third example, we take the situation opposite to the previous one: let it be necessary to simultaneously calibrate three devices with an accuracy ratio of σ ₁ : σ ₂ : σ ₃ ≈1: 3: 3, where the first device is again the main one, and the other two are additional but which this time is about three times worse in accuracy than the main one. In this situation, it is also possible to obtain satisfactory estimates of the main parameter σ ₁ , but already under a rather stringent condition n> 10100, which may require additional efforts to ensure the stationary errors of the calibrated devices throughout the measurement time, the duration of which depends on the data acquisition period Δt and can lie in the range from a few minutes to several hours (and wider).

It is easy to see that, in the conditions of the second example, the prototype method, although it can be formally applied, however, without guaranteed estimates of the reliability of the results. In the conditions of the first and third examples, the prototype method fundamentally does not allow application. Moreover, one should pay attention to the widely accepted opinion that it is impossible to certify the accuracy of a measuring device without necessarily relying on a more accurate (reference) device. However, the proposed method can successfully solve, including a similar problem.

The operability of the proposed calibration method has been tested in practice using simultaneous laboratory tests of three copies of the high-precision frequency meter CNT-90 (manufactured by Pendulum Instruments AB, Sweden) by repeatedly measuring the carrier frequency of various crystal oscillators (including low-stability, temperature-dependent and others factors). Obtained using the proposed method, reliable estimates of the standard deviation of random errors of the tested devices made it possible not only to confirm their compliance with their passport data, but also to perform their individual ranking by this indicator and choose the most accurate of the three devices. Thus, the verification of these instruments was actually performed (in terms of estimating the standard deviations of their random errors), however, they did not require expensive even more accurate reference (model) instruments.

Sources

1. RMG 29-2013 “GSI. Metrology. Basic terms and definitions. "

2. GOST R 8.674-2009 "GSI. General requirements for measuring instruments and technical systems and devices with measuring functions. "

3. MI 1317-2004 “GSI. Results and characteristics of measurement error. Presentation Forms. Ways to use when testing product samples and monitoring their parameters. ”

4. A.M. Yaglom. Introduction to the theory of stationary random functions. - UMN, 1952, volume 7, issue 5 (51), p. 3-168.

5. G. Kramer. Mathematical methods of statistics. - M.: Mir, ed. 2nd, eras., 1975 .-- 648 p.

6. www.vniims.ru/inst/gosreestr.html

7. Burdun G.D., Markov B.N. Fundamentals of Metrology. - M .: standards, 1975 .-- 336 s.

8. Astashenkov A.I., Nemchinov Yu.V., Lysenko V.G. Theory and practice of verification and calibration. - M .: Publishing house of standards, 1994. - 142 p.

9. Sergeev A.G., Krokhin V.V. Metrology: Textbook. manual for universities. - M .: Logos, 2001 .-- 408 p.

10. Malikov M.F. Fundamentals of Metrology. - M.: Publishing house Kommerpribor, 1949. - 480 p.

11. Rabinovich S.G. Measurement errors. - L .: Energy, 1978.- 262 p.

12. Mosteller F., Tukey J. Data Analysis and Regression (in 2 issues). Vol. 1. - M .: Finance and statistics, 1982. - 317 p.

Claims (1)

The calibration method of the device intended for measurement, including such devices as measuring instruments and technical systems and devices with measuring functions, used to evaluate the accuracy index of the calibrated device, where the mean square deviation of the random error component of the calibrated device assuming its dominance over the remaining components of the error, and carried out by comparing the discrete readings of the calibrated device and with discrete readings obtained simultaneously with them of another additional device of the same type in terms of measurable size with a calibrated device, characterized in that, firstly, at least two of the same type of additional devices are used simultaneously, for any of which it is not necessary to have an a priori estimate and / or the norms of its accuracy indicator, while all the devices involved in the calibration (including both the original calibrated device and all additional ones) are equal, including nothing the ratio of a priori estimates and / or norms (if any) of the accuracy indicators of these devices is regulated, and secondly, the purpose of the calibration is expanded to evaluate the adopted accuracy indicator for each of the simultaneously calibrated devices, and thirdly, that obtained by using the proposed method by appropriate mathematical processing of experimental data, estimates of the adopted accuracy indicator of each of these devices have the property of non-bias, and the reliability level of each of the obtained estimates NOC is fully characterized by analytical expressions for their dispersions, allowing, in particular, guaranteed to evaluate realistically achieved reliability levels for each of the resulting estimates.

Images